Page:Elektrische und Optische Erscheinungen (Lorentz) 056.jpg

 In an area that isn't too extended, we may also regard $$b_{x}$$, $$b_{y}$$, $$b_{z}$$ as constant, and thus regard the motion as a system of plane waves. The direction constants $$b'_{x}$$, $$b'_{y}$$, $$b'_{z}$$ of the wave normal are obviously to be determined from the condition

For $$\mathfrak{p}=0$$, $$b'_{x}$$, $$b'_{y}$$, $$b'_{z}$$ fall into $$b_{x}$$, $$b_{y}$$, $$b_{z}$$, and the waves are perpendicular to $$Q_0 P$$. This is not the case if the light source is moving. From (43) follows, that the waves are perpendicular to the line that connects P with that point at which the light source was at the moment, when the light was sent that reaches P at time t.

The law of Doppler.
§ 37. In a point that moves together with the luminous molecule — and thus also for an observer who shares the translation — the values of $$\mathfrak{d}_{x},...\mathfrak{H}_{x},...$$ are changing, as we have seen (§ 30), as often in unit time as it corresponds to the actual period of oscillation T of the ions.

We can also examine, with which frequency these values in a stationary point are changing their sign. This frequency causes the oscillation period for a stationary observer. The question can be solved immediately, if instead of x, y, z we introduce new coordinates $$\mathbf{x}$$, $$\mathbf{y}$$, $$\mathbf{z}$$, which refer to a stationary system of axes. If the two systems have the same directions of axes and the same origin at time t = 0, then

and by (42) for $$\mathfrak{d}_{x},...\mathfrak{H}_{x}, ...$$ we obtain expressions of the form

where